The offshore standard for submarine pipeline systems from Det Norsk Veritas is internationally recognized. This standard was developed based on the limit state design (LSD) criteria and using the load and resistance factors design (LRFD) principle. The combined loading criteria (CLC), applied to pipe members subjected to bending moment, axial force, and internal overpressure, essentially determines the design solutions for a pipeline. This requirement for all cross sections of a pipeline is described in the D505 clause of the DNV standard [1].

The existing CLC does not correctly consider the influence on the combined effect of the axial force for a pressurized pipeline. Equation 5.23 of D505, which describes CLC, includes a design effective axial force (*Sd*) that determines the global response of a pipeline. See clause 209. The axial force in a pipe wall (“true” force *Nd*) should be used instead of the effective axial force because equation 5.23 is based on parameters such as the yield stress and the tensile strength. Detailed discussion of “true” and effective axial forces was published previously [2].

The difference between the axial force in a pipe wall and the effective axial force is not only terminology. The quantity is significantly different in value as well as the sign (tensile or compressive). The sign of the axial force, i.e. whether the axial force is tensile or compressive, does not affect the CLC in equation 5.23. This contradicts the Von Mises or Tresca theories of combined (equivalent) stresses used for pressurized pipelines.

The method to determine the “true” steel wall axial force is outside the scope of this column. For a totally restrained and non restrained pipeline, the method is shown in equation 3. The method, which considers the soil resistance and the pipeline deflection for free span analysis is shown in equation 4.

The proposed method is based on classical Von Mises and Tresca combined (longitudinal and hoop) stresses theories, and on the plastic resistance criteria as the limit state. Von Mises and Tresca are used in US Codes B31.4 and B31.8 for offshore pipelines. DNV’s terminology, definitions, and symbols are used for the most part in this column.

The pipeline is modeled as the ring section beam under the bending moment (*M*) and the axial force in a pipe wall (*N*). The pipe material has different yield stresses at varying uniaxial tensile and compressive stresses due to the pressure hoop stress (σ*h*). Yield stresses in tensile (σ*y,t*) and in compressive (σ*y,c*) zones in the biaxial stress state (longitudinal and hoop [tangent] stresses) can be related to the yield stress (σ*y*) at the uniaxial tension by equation 1. Equation 1 describes the Tresca and Von Mises tensile and compressive yield surface by the parameter (ψ).

The pipe cross section plastic hinge, created by the bending moment and axial force, consists of the tensile and compressive zones of stress. These zones are separated by the neutral axis. The angle (β) defines the neutral axis location. This angle is measured from the extreme fiber of the tensile zone. Based on the above assumption, the axial force and the bending moment in the pipe ring cross section (medium radius of*r* and thickness of *t*) will be calculated from the static condition of equilibrium using equations 3 or 4, accordingly.

The size of the angle, that describes the location of the border between tensile and compressive areas, and the value of the plastic moment were found by integration as shown in 3 and 4. The results are presented in equation 5. In these equations,*N* and *M* are relative values (dimensionless parameters), the ratio of the axial force and the moment to the plastic axial force and to the plastic moment resistances, accordingly to equation 6. Note that the plastic resistances (*Sp* and *Mp* values) in equation 6 match those in DNV.

Equation 5, which describes the relationship between the bending moment, the axial force in pipe wall, and the overpressure at limit state of pipe section, represents the combined load criteria. The accompanying graph illustrates the relationships described by equation 5. This graph plots the maximum relative bending moment (*M* ) versus the relative axial force in pipe wall (*N*) for different values of the relative hoop stress (σ*h*), denoted in the graph as *S_h*.

As this graphs shows, the allowable bending moment is dependent on the value of the axial force as well as on the sign of the force. The internal pressure increasing also reduces the allowable moment. Dotted lines represent the DNV combined criteria per 5.23. The solution agrees with the DNV solution only when the pressure equals zero. In all other cases, the result is very different, especially when the axial force is compressive.

#### Application to design

Equation 5 encompasses the theoretical solution of the combined loading criteria for pipe members subjected to bending moment, axial force, and internal overpressure. For application in design, the equations were rewritten in the form of equation 7 to consider the design factors. Since DNV is based on LRFD consideration, relative parameters of loads and resistances used in equation 7 should be calculated in accordance with equations 8 - 10. The symbols and their definitions in these equations are identical to those in DNV.

Thus, the proposed methodology is based on classical theory of Von Mises and Tresca on combined stresses as well as the DNV philosophy of design loads factors, design resistances, and criteria of limit state design to allow for the safely design of offshore pipelines, especially HP/HT pipelines.

**Alexander Aynbinder**

WorleyParsons, Houston

#### References:

1. Offshore Standard DNV-OS-F101 “Submarine Pipeline System”, p.166, 2003

2. O. Fyrileiv and L. Collberg “Influence of pressure in pipeline design – effective axial force”, OMAE2005-67502, p. 8

3. DNV-RP-F105 “Free Spanning Pipelines” p. 45, 2006

4. A. Aynbinder “Method determines axial force in subsea pipelines” O&GJ, p 66-71, June 18, 2007

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